Some remarks on quasi-variational inequalities and the associated impulsive control problem
Benoı̂t Perthame · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 1985
We study Quasi-Variational Inequalities: \tag{1} \begin{cases} \mathrm{Max}(\mathrm{A}u−f,u−\mathrm{M}u) = 0, \\ u\mid_{∂\mathrm{\Omega }} = \varphi , \end{cases} where \tag{2} \mathrm{M}u = k + \inf \limits_{\stackrel{\xi ≧0}{x + \xi \in \:\bar{\mathrm{\Omega}}}} \left\{c_{0}(\xi ) + u(x + \xi )\right\}. In general, (1) has no solution, we prove here that (1) has a unique maximum subsolution that we caracterize. Then we compare the implicit obstacle (2) and the obstacle: \def\infess#1{\inf\limits_{#1}\hspace*{-8pt}\mathrm{ess}} \mathrm{M}_+ u = k + \infess{\stackrel{\xi ≧0}{x + \xi \in \:\mathrm{\Omega}}} \left\{c_{0}(\xi ) + u(x + \xi )\right\} and we finally show that, under general assumptions, the solution of (1) is Holder continuous. Résumé Nous étudions les Inéquations Quasi-Variationnelles : \tag{1} \begin{cases} \mathrm{Max}(\mathrm{A}u−f,u−\mathrm{M}u) = 0 &\text{dans}\:\mathrm{\Omega }\\ u\mid_{∂\mathrm{\Omega }} = \varphi , \end{cases} où \tag{2} \mathrm{M}u = k + \inf \limits_{\stackrel{\xi ≧0}{x + \xi \in \:\bar{\mathrm{\Omega}}}} \left\{c_{0}(\xi ) + u(x + \xi )\right\}. En général (1) n’a pas de solution, nous montrons ici que (1) admet une unique sous-solution maximale que nous caractérisons. Nous comparons ensuite l’obstacle implicite (2) et l’obstacle : \def\infess#1{\inf\limits_{#1}\hspace*{-8pt}\mathrm{ess}} \mathrm{M}_+ u = k + \infess{\stackrel{\xi ≧0}{x + \xi \in \:\mathrm{\Omega}}} \left\{c_{0}(\xi ) + u(x + \xi )\right\} et nous finissons par montrer que, sous des hypothèses générales, la solution de (1) est hölderienne.